Three procedures, one panel

A drawing of a tiled floor, receding to a point. Somebody made it, and they made it by a procedure — Alberti’s lateral section, the distance point, a measuring line, a photograph. Which one?

The question sounds like connoisseurship and it is arithmetic. A procedure is a rule for putting marks on a panel, the marks are still there, and if two rules put marks in different places then the places say which rule ran.

This collection has already established that they do not.

The three routes agree to the last bit #

Alberti draws a pavement runs the costruzione legittima of 1435 as a construction rather than as a formula: a side elevation beside the panel, the eye at its height above the ground, a ray from the eye to each braccio mark, and the transversal read where the ray crosses the panel. The distance point is the viewing distance runs the other classical recipe: orthogonals to the centric point, one further point on the horizon at the viewing distance, and the diagonal from the near corner crossing each orthogonal in turn.

Neither evaluates a closed form. Both draw lines and intersect them, with the same two functions the rest of this site uses on drawn pictures. And both agree with a pinhole camera built to the panel’s own focal length, to the arithmetic floor.

The insets are the only reason there is anything to see. Remove them and the three sets of transversals are one set of transversals.

So the first thing to say about attribution is a refusal, and it is complete: a correctly executed drawing carries no record of which correct procedure executed it. Not “very little record”. None.

What a reader actually holds #

Everything below is written for somebody with the drawing and nothing else — no camera, no focal length, no statement of what the panel was built to. That is the ordinary situation, and it is the situation the whole metrology field is written for as well: a height, out of one photograph takes the same view of what is available.

What is available is a set of marks. On a pavement of evenly divided braccia the marks that matter are the transversals — the images of the lines where each row of tiles ends — and they lie on one line up the panel.

The reader’s instrument is the invariant. Four collinear points have a cross-ratio; a projection preserves it; four evenly spaced collinear points have the cross-ratio 4/3, computed from the parameters 0, 1, 2 and 3 and from nothing else. So the transversals of an evenly divided pavement have to give 4/3, whatever the panel’s focal length, wherever its horizon is, and at whatever scale it was drawn.

That the test needs no horizon is worth dwelling on. Four lines have a cross-ratio establishes the invariant on a pencil; an angle is a cross-ratio is the same fact used to get an angle back out of a picture. Here it is used the other way round: not to recover something, but to ask whether a set of marks is consistent with having been projected at all.

Run it on the three drawings above and every quadruple returns 4/3 to fifteen digits. The instrument works and it reports nothing, because there is nothing to report.

The difference is where the hand goes #

If the answers coincide, the procedures can only be told apart by their mistakes — and their mistakes are as different as their answers are alike, because each procedure has exactly one step a person performs.

Alberti’s section has the draughtsman mark each braccio along the section’s ground line, measuring each one from the panel. A slip on the fourth mark moves the fourth transversal and leaves the rest where they were.

The distance point has the draughtsman put one mark on the horizon. A slip in it changes the panel’s focal length and nothing else.

A measuring line walked with dividers has the draughtsman step each length off from the last. A slip on the fourth step moves the fourth transversal and every one after it.

A photograph has no hand step at all. What it has instead is a lens.

Four hand steps, four different kinds of trace. That is the whole of the row, and the rest of these essays are the four traces measured.

How much a slip moves the number #

Before the traces are measured it is worth knowing what size of slip the instrument can see, because a test that reports 4/3 to fifteen digits on a perfect drawing may still report 4/3 on a badly executed one.

Displace the third of four evenly spaced ground marks by $\varepsilon$ braccia. The four world parameters become $0, 1, 2+\varepsilon, 3$, and their cross-ratio is

$$\frac{(2+\varepsilon)\cdot 2}{(1+\varepsilon)\cdot 3} \;=\; \frac{4}{3} - \frac{2}{3}\varepsilon + O(\varepsilon^{2}),$$

so a slip of one part in a hundred of a braccio moves the cross-ratio by about 0.0067, or half a per cent of its own value. The invariant is not merely a yes-or-no test; it is linear in the mistake, with a coefficient of two thirds.

What decides whether that is detectable is the other end of the chain. A cross-ratio read off a drawing is only as good as the smallest of the three gaps between its four marks, since that gap is the denominator the reading error divides into — and on a pavement the gaps close up as the square of the count. So the near quadruples are sensitive and the far ones are blind, and a slip in the fifth or sixth mark of a receding floor is invisible for the same reason the floor’s own bands are: there is nothing left to measure it against.

There is one more piece of arithmetic that decides what the row can do at all. A set of $n$ collinear marks carries $n-3$ independent projective invariants, so a pavement with seven transversals offers four independent numbers rather than the thirty-five its quadruples suggest. Four numbers is enough to localise a mistake, and that is exactly the distinction the procedures need: an isolated slip in the $k$-th mark perturbs only the invariants whose quadruples contain it, while a cumulative slip in a stepped-off sequence perturbs every invariant from $k$ onward. One shows up as a spike and the other as a step.

Which is the essay’s thesis with a shape attached. The procedures are indistinguishable in what they get right and distinguishable in the pattern of what they get wrong — not the size of the error, which says only how careful the hand was, but where in the sequence the deviations start and whether they stop.

The one that leaves nothing #

One of the four is different in kind, and it is the one the site’s own premise is about.

A slip in placing the distance point does not produce a worse drawing. It produces another exactly correct drawing, of a room with different proportions, seen from a different distance. The transversals move — visibly, by pixels — and the cross-ratio stays at 4/3 to the arithmetic floor, because a correct perspective is what the construction still is.

So the reader’s instrument, which is the strongest test a single picture admits, cannot see the one error that changes what the drawing means. The point to stand at is the number the slip moves, and standing in the wrong place is what it costs a reader who does not know it moved.

This is the shape Brunelleschi drilled a hole in his panel is about, arrived at from the other end. He fixed the viewing distance with a hole because nothing in the picture fixes it; five centuries later, nothing in the picture fixes it.

What the other three leave #

The other three procedures do leave something, and it is a shape rather than a number.

Subtract the best correct perspective from a drawing and what is left is a residual, mark by mark. A scatter, a wander, or a smooth curve — and which of the three it is depends only on how the hand’s errors combined, not on how large they were.

The statistic that separates them is the residual’s roughness, and it is arithmetic rather than a threshold somebody picked: the second difference of independent errors has six times their variance, so an independent residual’s roughness is the square root of six. Stepped, or measured from the zero is where that is measured against the count, and it holds.

The count that says when there is anything to read #

Before any of this, a drawing has to have enough marks in it to disagree with anything, and the count is exact.

A reader fitting a correct perspective has three numbers to choose: where the horizon is, and the two that fix the projective map of the receding line. Each transversal is one equation. So a pavement of three transversals is fitted exactly whatever its transversals are — every set of three is a correct perspective of something — and the fourth is the first that can fail.

The same count arrives from the cross-ratio side: a pavement of n transversals contains n − 3 consecutive quadruples, which is the same number. That is not a coincidence and it is the ordinary situation for a projective test: what one picture of a plane determines counts the same three degrees of freedom from the other direction.

The practical reading is blunt. A three-tile pavement cannot be attributed to anything, and no amount of care in measuring it changes that. A great many paintings have three-tile pavements.

The rule that is not a projection #

There is a fifth procedure, and it is the one the wrong field exists for: each gap a fixed fraction of the last, which is what art classes teach for spacing receding boards. It is not a projection of anything, so the reader’s instrument ought to convict it at once.

It does convict it, arithmetically. What that conviction is worth in pixels is a different question with a surprising answer, and it is the reason this row needed a second instrument.

The sixth route, which commits to nothing #

There is one further way to space a pavement and it belongs here because it is the limiting case of the whole argument.

The bay repeated by a straightedge lays off equal depths using only the horizon and a ruler: the diagonal of one bay, extended, gives the far edge of the next, and the construction repeats indefinitely without measuring anything. Seven is not a power of two is the same machinery asked for a division rather than a repetition, and it is exact for every rational fraction.

What makes it the limiting case is that it has no free parameter to slip. Alberti’s section chooses a viewing distance and so does the distance point; the straightedge route chooses nothing. Hand it the first bay and the horizon and every subsequent transversal is determined, so a draughtsman using it cannot misplace a distance point because there is no distance point to place.

That does not make it unattributable in a different way. Its hand step is the ruler laid across two drawn points, so its errors are independent in exactly the sense Alberti’s are, and a drawing made this way is read as a measured hand — correctly. What it means is that the viewing distance the drawing implies is inherited from the first bay rather than chosen, which relocates the free parameter into the one measurement the draughtsman did make rather than removing it.

Why the agreement is exact rather than close #

It is worth saying why the three routes coincide, because “they agree to fifteen digits” invites the reading that they are three good approximations to one another and they are not.

A ground point k braccia beyond the picture plane sits at a depth proportional to 1 + k·unit/focal, and its image is the horizon plus the rise divided by that. Alberti’s section computes it by intersecting a drawn ray with a drawn panel; the distance point computes it by intersecting a drawn diagonal with a drawn orthogonal; a camera computes it by dividing by depth. All three are the same projective map of the same line, written in three notations, so they are not close — they are equal, and the fifteen digits are the floor of the arithmetic rather than the accuracy of the agreement.

That is also why the coincidence is not fragile. Change the panel, change the focal length, change the number of braccia, and the three still agree, because nothing about the setting was doing the work. Three constructions, one map is the general statement of it in the foundations field, and this row is one instance of that statement with a question about authorship attached.

Reading the panel and not only the pavement #

The second instrument is the panel’s own horizon.

A panel shows where its orthogonals meet: it is a mark on the page, and a reader with a straightedge can find it. The transversals separately imply a horizon, because their ratios only fall on a line for one value of it. On a correctly constructed panel the two are the same point.

Admitting the horizon is what turns a reading that misses two of the five into one that misses one. The one it still misses is the distance point, and it always will.

What this row can and cannot deliver #

Set out plainly, because the literature on attributing perspective constructions is confident and this is not.

A drawing can be shown not to be a projection at all. That is the cross-ratio, it needs four transversals, and it is decisive.

A drawing can be shown to disagree with its own panel. That is the implied horizon against the drawn one, it needs no statistics, and it convicts the taught spacing rule at a glance.

A drawing’s kind of hand error can be read. Independent, cumulative, or a rule — from the residual’s roughness, in the ensemble, with an honest failure rate on any one panel.

A drawing’s procedure cannot be named. Not the distance point against Alberti’s section, and not a well-executed anything against a well-executed anything else. What a panel says about its maker is where that is stated with its confusion matrix, and the matrix is most of the content.

Why it is worth a ladder #

Three reasons, and the second is the one that changed how this collection reads the classical material.

The negative is the useful half. A claim that a panel was laid out by the distance point because the pavement is regular is a claim with nothing behind it. Knowing exactly which claims of that shape are empty is worth more than a method that occasionally works.

It relocates the free parameter. The viewing field has argued from the beginning that a construction chooses where the reader stands and does not say so. This row shows that the drawing does not say so either — the choice leaves no trace anywhere, in a picture whose every other property is checkable.

And it puts a number on the classical drawings that pass. A rule that has been taught for five centuries and is not a projection produces panels that read as correct perspectives up to a stated size. That size is nine braccia, and it is larger than most of the pavements anybody painted.

The short version #

Three correct procedures put every transversal at the same pixel, so a correctly executed drawing says nothing about which one made it. What it can say is whether it is a projection at all — the cross-ratio of four consecutive transversals is 4/3, and four is the fewest that can disagree — and whether the pavement agrees with the panel it sits on.

The one error the reading cannot see is the one that matters: a misplaced distance point produces another exact drawing, correct from somewhere else entirely, and no test of the picture will ever find it.